A function can be linear.
An operator can be linear.
We will introduce the idea by showing what this means for both of them and we believe from that you can infer the general idea of linearity.
Linear Function
f(u+v) = f(u) + f(v)
f(ax) = af(x)
If u,v,x are numbers, and a is also a number, we can illustrate the first identity with
f(7) = f(5) + f(2)
and we can illustrate the second identith with
f(15) = 3*f(5)
Appendix A
Sometimes the two rules are merged into one rule:
f(ax + by) = af(x) + bf(y)
If necessary, observe the change in two steps:
f(ax + by)
f(ax) + f(by)
af(x) + bf(y)
Appendix B
- f(x) = mx
- This function is linear
- f(x) = mx + b
- This function is not linear except for the trivial case where b equals zero
- This function is affine
It might be surprising to learn the dean a straight line is not enough to make a function linear. In order to be linear, that’s straight line must go through the origin.
Appendix C
The tangent space varies from
point to point, unless all functions yi
(x1,…,xn) are linear.
We found this and we are going to run with it.
Take the function y=kx
The slope of every point on that function is k.
For a one variable function, the tangent space is just a line.
For a two variable function, the tangent space is a plane. Another way of explaining it–that plane contains a possible tangent lines through that point.
Vectoreverything 